An Existence Result for a Generalized Quasilinear Schrödinger Equation with Nonlocal Term

Abstract

In this paper, we consider the following generalized quasilinear Schrödinger equation with nonlocal term $-\mathrm{div}\left(g^2\left(u\right)\nabla u\right)+g\left(u\right)g^{\prime }\left(u\right){\left|\nabla u\right|}^2+V\left(x\right)u=\lambda \left[{\left|x\right|}^{-\mu }\ast {\left|u\right|}^p\right]{\left|u\right|}^{p-2}u,x\in {ℝ}^N,$ where $N\ge 3$ , $g:ℝ\to {ℝ}^{+}$ is a $C^1$ even function, $g\left(0\right)=1$ , $g^{\prime }\left(s\right)\ge 0$ is for all $s\ge 0$ , $\underset{\mid s\mid \to +\infty }{\lim }g\left(s\right)/{\left|s\right|}^{\alpha -1}≔\beta >0$ is for some $\alpha >1$ , and $\left(\alpha -1\right)g\left(s\right)\ge g^{\prime }\left(s\right)s$ is for all $s\ge 0$ , $2\alpha \le p\le 2\alpha \left(N-\mu \right)/N-2$ , and $0<\mu <N$ . We prove that the equation admits a solution by using a constrained minimization argument.